HiDeNN-FEM: a seamless machine learning approach to nonlinear finite element analysis
HiDeNN-FEM: a seamless machine learning approach to nonlinear finite element analysis
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DOI：
10.1007/s00466-023-02293-z
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发表时间：
2023-04
影响因子：
4.1
通讯作者：
Yingjian Liu;Chanwook Park;Ye Lu;S. Mojumder;Wing Kam Liu;Dong Qian
中科院分区：
文献类型：
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作者：
Yingjian Liu;Chanwook Park;Ye Lu;S. Mojumder;Wing Kam Liu;Dong Qian
分层深度学习神经网络（HiDeNN）（Zhang et al. Computational Mechanics，67：207-230）提供了一种构建数值近似的系统方法，可以将其纳入各种偏微分方程（PDE）和/或常微分方程（ODE）求解器中。提出了一种基于HiDeNN逼近的非线性有限元框架（Nonlinear HiDeNN-FEM）。这是通过采用结构化深度神经网络的三个基本构建块实现的：（1）偏导数算子块，其执行形状函数相对于元素坐标的微分;（2）r自适应块，其改善局部和全局收敛特性;以及（3）材料导数块，其评估形状函数的材料导数。虽然这些构建块可以应用于任何元素，但具体实现以1D和2D形式呈现，以说明深度学习神经网络的应用。两步优化方案的进一步发展，以允许的r-自适应能力和易于集成与任何现有的有限元求解器。二维和三维的数值算例表明，所提出的具有r-自适应性的非线性HiDeNN-FEM比常规FEM具有更高的精度。它还可以显著减少元素变形并抑制沙漏模式。
The hierarchical deep-learning neural network (HiDeNN) (Zhang et al. Computational Mechanics, 67:207–230) provides a systematic approach to constructing numerical approximations that can be incorporated into a wide variety of Partial differential equations (PDE) and/or Ordinary differential equations (ODE) solvers. This paper presents a framework of the nonlinear finite element based on HiDeNN approximation (nonlinear HiDeNN-FEM). This is enabled by three basic building blocks employing structured deep neural networks: (1) A partial derivative operator block that performs the differentiation of the shape functions with respect to the element coordinates, (2) An r-adaptivity block that improves the local and global convergence properties and (3) A materials derivative block that evaluates the material derivatives of the shape function. While these building blocks can be applied to any element, specific implementations are presented in 1D and 2D to illustrate the application of the deep learning neural network. Two-step optimization schemes are further developed to allow for the capabilities of r-adaptivity and easy integration with any existing FE solver. Numerical examples of 2D and 3D demonstrate that the proposed nonlinear HiDeNN-FEM with r-adaptivity provides much higher accuracy than regular FEM. It also significantly reduces element distortion and suppresses the hourglass mode.